Approximation of unbounded functions and applications to representations of semigroups

AbstractThe purpose of this paper is to study the approximation of functions in m variables and its application to semigroup representation. First, two Bohman-Korovkin-type theorems are established for the respective approximations of unbounded, operator-valued and real-valued functions with noncompact supports in Rm. Then we investigate several approximation operators; some of them are generalizations (to m dimensions) of well-known linear positive operators and some are apparently new. Finally, through these operators, the first approximation theorem provides a unified approach to a whole set of representation formulas for m-parameter (C0)-semigroups of operators; special cases include well-known formulas due to Hille, Phillips, Widder, Kendall and Chung, as well as some new ones